this paper we have focused on homoclinic bifurcation in
a second order nonlinear differential equation.
Bifurcation theory attempts to provide a systematic classification of the
sudden changes in the qualitative behaviour of dynamical systems. A
bifurcation occurs when a small smooth change made to the parameter values (the
bifurcation parameters) of a system causes a sudden qualitative change in its
behaviour. Bifurcations are broadly classified into two types- local and
global. Local bifurcation is associated
with equilibria or cycles. Homoclinic bifurcation belongs to the global bifurcation
category which deals with bifurcation events that involve larger scale
behaviour in state space. A bifurcation which is characterized by the
presence of trajectory connecting equilibrium with itself is called homoclinic
bifurcation. Roughly speaking, a
homoclinic orbit is an orbit of a mapping or differential equation which is
both forward and backward asymptotic to a periodic orbit which satisfies a
certain non-degeneracy condition called ‘‘hyperbolicity”.
The Melnikov method which uses Melnikov distance function
provides a measure of the distance between a
stable and unstable manifold. This method is used in our investigation.
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